Talks in Maths: Visualizing Repetition in Text and

the Fractal Nature of Lyrical Verse.

A thesis submitted in partial fulfillment of the requirements for the degree of

Master of Science in Computer Science & Engineering

by

William C. Kurt

Dr. Frederick C. Harris, Jr., Thesis Advisor

August, 2014

We recommend that the thesis prepared under our supervision by

WILLIAM C. KURT

Entitled

Talks In Maths: Visualizing Repetition in Text and the Fractal Nature of Lyrical Verse.

be accepted in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

Dr. Frederick C. Harris, Jr., Advisor

Dr. Richard Kelley, Committee Member

Mr. Larry Dailey, Graduate School Representative

David W. Zeh, Ph.D., Dean, Graduate School

August, 2014

THE GRADUATE SCHOOL

i

Abstract

Using the techniques of Natural Language Processing (NLP), the repetition in the

structure of lyrical verse (song lyrics and poetry) can be visualized by comparing the cosine

similarity between each line in a given document. This visualization allows novel insight into the

structure of repetition in lyrical verse, allows for the ability to see how this repetition can shape

the structure of a poem or song, as well as providing a deeper understanding of how the

repetition in lyrical works is generated. The key insight arrived at within this work, made clear

through this visualization technique, is that the structure of dense repetition in lyrical verse is

often fractal in nature. The fractal nature of song lyrics is explored and measured using the

mass dimension. Ultimately this leads to a deep insight into the way fractals drive the aesthetic

properties of poetry and song lyrics.

ii

Dedication For Lisa and Archer

iii

Acknowledgements

I would like to thank my committee members, Dr. Frederick Harris, Jr., Mr. Larry Dailey,

and Dr. Richard Kelley. Extra thanks go to Dr. Harris for serving as a fantastic advisor and Dr.

Kelley for pushing me past the easy stopping point in order to find something truly exciting.

A tremendous thank you is needed for my wife Lisa for being so patient and helpful

during the many nights I spent locked in an office printing out crazy visualizations and putting

this work together. An especially large thank you must go to my son Archer, who at 3 years old

stayed up late with me asking to learn about self-similarity, drawing his own Sierpinksi Gasket,

and without whom I would never have made the connection between Dr. Seuss and fractal

geometry.

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Contents Abstract i Dedication ii Acknowledgments iii Table of Contents iv List of Figures vi 1 Introduction 1 2 Background 3 2.1 Natural Language Processing ....................................................................................... 3 2.1.1 Representation of Text ................................................................................... 5 2.1.2 Determining Similarity Between Texts ........................................................... 6 2.2 Fractals .......................................................................................................................... 7 2.2.1 Fractals in Nature ......................................................................................... 10 2.2.2 Measuring Fractals ....................................................................................... 12 2.2.3 Measuring Fractals in Nature ....................................................................... 14 2.3 Visualizing Text ............................................................................................................ 17 2.3.1 Other Examples of Text Visualization .......................................................... 20 2.3.2 Visualizing Poetry and Song Lyrics ............................................................. 22 3 Visualizing the Structure of Repetition in Lyrical Verse 28 3.1 Overview ...................................................................................................................... 28 3.2 Visualizing Repetition .................................................................................................. 28 3.3 Why Visualize? ............................................................................................................ 29 3.4 Close Reading ............................................................................................................. 29 3.5 Mathematics of Aesthetics .......................................................................................... 31 4 Implementation 33 4.1 Preprocessing .............................................................................................................. 33 4.2 NLP .............................................................................................................................. 34 4.2.1 NLP Preprocessing ...................................................................................... 34 4.2.2 Common NLP Preprocessing Avoided ........................................................ 34 4.2.3 Vectorization ................................................................................................. 35 4.2.4 Sparse Representation .................................................................................. 36 4.3 Visualization ................................................................................................................ 36 4.4 Computing Fractal Dimension .................................................................................... 38 4.4.1 Visualizing Fractal Dimension ...................................................................... 39

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5 Results 41 5.1 Visualization - Simple Applications ............................................................................ 41 5.1.1 Visualizing the Thunder - Visualization and Close Reading........................ 41 5.1.2 Understanding the Structure of Popular Songs ........................................... 44 5.2 The Fractal Nature of Lyrical Verse ........................................................................... 48 5.2.1 Green Eggs and Ham and Self-Similarity .................................................... 48 5.2.2 Searching for Fractals in Song Lyrics .......................................................... 54 5.2.3 Song Lyrics and Cantor Dust ....................................................................... 63 5.3 Measuring Fractals in Lyrical Verse ........................................................................... 66 5.4 Interpreting the Fractal Dimension of Lyrics .............................................................. 68 6 Conclusion and Future Work 71 6.1 Conclusion .................................................................................................................. 71 6.2 Future Work ................................................................................................................ 73 6.2.1 Modeling Others Aspects of Repetition ....................................................... 73 6.2.2 Website for Aggregation ............................................................................... 73 6.2.3 Generative Models ....................................................................................... 74 6.2.4 Generalization ............................................................................................. 74 Bibliography 75

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List of Figures 2.1 Measuring coast of Britain 200km measuring stick [53] ......................................................... 7 2.2 Measuring coast of Britain 50km measuring stick [54] ........................................................... 8 2.3 Brownian motion at 100, 1,000, 10,000, and 1,000,000 samples .......................................... 9 2.4 Cantor set [57] ....................................................................................................................... 10 2.5 Sierpinski gasket [69]............................................................................................................. 10 2.6 Romanesco cauliflower [58] .................................................................................................. 11 2.7 Doodles of Archer Kurt (age 3) compared to Levy flight ....................................................... 11 2.8 Approximating circumference of circle .................................................................................. 13 2.9 Coastline of Norway [66] ....................................................................................................... 14 2.10 Estimating the box counting dimension of the coast of Britain [60] .................................... 15 2.11 Goose down [44] .................................................................................................................. 16 2.12 3-D Lichtenberg figure in acrylic [67] ................................................................................... 16 2.13 Background word cloud ....................................................................................................... 18 2.14 Phrase net  for  “Pride  and  Prejudice”  [36] ............................................................................ 19 2.15 Word tree for  “I  Have  a  Dream”  [70] .................................................................................... 20 2.16 Image of latent semantic similarity [79] ............................................................................... 21 2.17  “Writing  Without  Words”  [38] ............................................................................................... 22 2.18 Visualizing Sonnet [1] .......................................................................................................... 23 2.19 Screenshot from POEM Viewer .......................................................................................... 23 2.20  Screenshot  of  Group  Lab’s  lyric  visualization ..................................................................... 24 2.21 J.Oh lyrics visualization [30] ................................................................................................ 25 2.22 Figure 1 from OK Go booklet [31] ....................................................................................... 26 2.23 Figure 2 from OK Go booklet [31] ....................................................................................... 26 3.1  Line  similarity  of  Blake’s  “Divine  Image” ............................................................................... 30 3.2 Songs of Innocence - Divine Image [61] ............................................................................... 31 4.1 Line similarity for  Vampire  Weekend’s  “Ya  Hey”  [23] ........................................................... 38 4.2 Calculating the mass dimension ............................................................................................ 40 4.3 Log-log plot for approximating mass dimension ................................................................... 40 5.1  Line  similarity  of  T.S  Eliot’s  “The  Wasteland  - what  the  thunder  said” ................................. 42 5.2 Close-up  of  “DA”  section ....................................................................................................... 43 5.3  Line  similarity  for  Vampire  Weekend’s  “Walcott” .................................................................. 44 5.4  Line  similarity  for  all  songs  on  the  album  “Modern  Vampires  of  the  City” ............................ 46 5.5  Line  similarity  for  Radiohead’s  “Karma  Police” ..................................................................... 47 5.6  Line  similarity  for  “Burnt  Norton  III” ........................................................................................ 47 5.7  Line  similarity  for  “East  Coker  III” .......................................................................................... 48 5.8  Line  similarity  for  “Green  Eggs  and  Ham” ............................................................................. 50 5.9  Zoomed  in  segment  of  “Green  Eggs  and  Ham” .................................................................... 51 5.10 Highlighted self-similarity  in  “Green  Eggs  and  Ham” .......................................................... 52

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5.11 Line  similarity  in  “Fox  in  Sox”  with  self-similarity highlighted .............................................. 53 5.12 Line  similarity  in  “Hop  on  Pop”  with  self-similarity highlighted ............................................ 54 5.13 Line  similarity  for  Tame  Impala’s  “Why  Won’t  They  Talk  to  Me” ........................................ 55 5.14 Line  similarity  for  Tame  Impala’s  “Feels  Like  We  Only  Go  Backwards” ............................. 56 5.15 Line  similarity  for  Vampire  Weekend’s  “Obvious  Bicycle” .................................................. 57 5.16 Vampire  Weekend’s  “Obvious  Bicycle”  with  self-similarity highlighted............................... 58 5.17 Line similarity  for  Radiohead’s  “Subterranean  Homesick  Alien” ........................................ 59 5.18 Line  similarity  for  Radiohead’s  “The  Tourist” ...................................................................... 59 5.19 Line  similarity  for  Radiohead’s  “Bullet  proof  ….  I  wish  I  was” ............................................. 60 5.20 Line  similarity  for  Radiohead’s  “High  and  Dry” .................................................................... 60 5.21 Line similarity  for  Vampire  Weekend’s  “Hudson” ................................................................ 61 5.22 Line  similarity  for  Lana  Del  Rey’s  “Million  Dollar  Man” ....................................................... 61 5.23 Line  similarity  for  Lana  Del  Rey’s  “Radio” ........................................................................... 62 5.24 Line  similarity  for  Tame  Impala’s  “Feels  Like  We  Only  Go  Backwards” ............................. 62 5.25 Line similarity for Lana  Del  Rey’s  “Diet  Mountain  Dew”...................................................... 63 5.26 Line  similarity  for  Radiohead’s  “Climb  Up  the  Walls” .......................................................... 64 5.27 Line  similarity  for  Vampire  Weekend’s  “one” ...................................................................... 64 5.28 Line  similarity  for  Radiohead’s  “Ripcord” ............................................................................ 65 5.29 Cantor dust [55] ................................................................................................................... 65 5.30 (a) Mass dimension Tame  Impala  “It  is  not  meant  to  be” ................................................... 67 5.30 (b) Mass  dimension  Vampire  Weekend  “Cousins” ............................................................. 67 5.30 (c) Mass  dimension  Lana  Del  Rey  “Radio” ......................................................................... 67 5.31 Mass  dimension  for  “The  Technological  Society” ............................................................... 69 5.32 Mass  dimension  for  “The  Technological  Society”  decreased threshold ............................. 69 6.1 Mandelbrot set [63] ................................................................................................................ 72

1

Chapter 1 Introduction “What  immortal  hand  or  eye Dare  frame  thy  fearful  symmetry?” -- William Blake, The Tiger

Repetition is one of the most fundamental aspects of the aesthetic experience. From the

repeating chorus in a popular song to the allusion to past masters in great works of art. Whether

echoing the previous line or generations long past similarity, symmetry, and synchronicity (not to

mention alliteration) are all core component which can lead to that chilling moment of the viewer

encountering a moment of beauty, and all these things are forms of repetition. And yet repetition

in art must remain elusive. Few things could be further from beauty than an endlessly skipping

record, and yet that same exact record, sampled and re-sampled in music can create engaging

experiences bordering on the hypnotic.

In our present undertaking we seek to uncover the work of repetition in lyrical verse,

exploring both poetry and lyrics in popular music. Even given this specificity we focus

exclusively on the repetition created by the similarity in word composition of individual lines. In

doing this we are not only able to uncover novel observations about individual works, but craft a

larger understanding of how repetition works at a larger scale, and gain a peek at precisely how

repetition is able to craft the aesthetic experience while hiding its true structure.

We extract the complicated structure of repetition from lyrical verse by creating a

compact visual representation that allows the observer to view all the relationships between

each line while still preserving the sequential structure of the text. This is achieved by taking a

vector representation of a given text, and then plotting out the similarity between each line and

all of the other lines individually. The end result is an n x n grid where n is the number of lines in

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the work, and each cell in this grid is color coded to represent the degree of similarity between

the two lines.

This visualization offers a wide range of insights into both the individual works and

patterns that stretch across multiple works, artists, and genres. The most simple application is in

aiding close reading by allowing the viewer to easily see how smaller parts of the work factor

into the larger structure of repetition in the overall piece. One of the most surprising insights

provided by these visualizations is that, in works with particularly dense structure of repetition,

the structure of repetition exhibits clear self-similarity. This observation of self-similarity then

leads to the exploration of the fractal nature of many works of lyrical verse, particularly in the

lyrics of popular songs. From this we are able to arrive at a methodology to calculate the fractal

dimension of these works.

Far from a mere novelty, understanding the fractal dimension of popular songs and

certain poems leads to an understanding of the mathematical mechanics responsible for

generating the aesthetic properties that separate these lyrical verses from prose. In the end we

arrive at a stronger position to analyze and understand the very nature of the repetition that

makes lyrical verse so aesthetically pleasing. This in turn leads us one step closer to having a

sophisticated mathematical understanding of the nature of the aesthetic.

3

Chapter 2 Background “It  feels  like  we  only  go  backwards  baby, Every  part  of  me  says  go  ahead”   -- Tame Impala, Feels Like We Only Go Backwards

2.1 Natural Language Processing

A core component of our work requires at least a basic understanding of Natural

Language Processing (NLP). NLP is required for us to transform language (in our case

specifically text) into a format that is understandable by machines and easily manipulated by

mathematical models. In NLP we refer to a body of text as a corpus [27]. What encompasses a

corpus can vary depending on exactly what types of text you are working with. For example a

collection of all English language texts could appropriately considered a corpus, while in our

case, a corpus will typically consist of a collection of lines of lyrical verse (from either poems or

song lyrics), sometime across multiple works.

From the idea of a corpus we must then move forward and understand how exactly we

are going to start breaking up the individual texts. At the most basic level we break the text into

a sequence of tokens [27, 28]. Most commonly one can think of a token as an instance of a

word; however, as we shall soon see this is not always the case. As an example the following

sentence:

“And  drank  coffee,  and  talked  for  an  hour.”

can be tokenized to the following array of words:

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{and, drank, coffee, and, talked, for, an, hour}

Next we look at the types and terms [27, 28]. While tokens are simply the instance of the word,

the type refers to the similarity shared by multiple tokens. This can be thought of as the

vocabulary of the text. Term simply refers to the types that are known in our corpus. For our

purposes we will never bother with text outside of a given corpus so we do not have to worry

about encountering types which are not terms, and will from here on exclusively refer to the

terms in the corpus. The terms in the previous sentence are:

{and, drank, coffee, talked, for, an, hour}

Notice  that  ‘and’  does  not  appear  twice  as  it  does  when  we  looked  only  at  words.

The next challenge for NLP is understanding how to represent word order (if it is

represented at all). The most common approach to this is the n-gram model. An n-gram

considers the preceding words in a sequence and then assumes the text possesses the Markov

property [27, 28, 64]. The Markov property simply states that [27, 28]:

P(X( )   ⊥   X( : )  |  X( ))

Which can be stated that the probability of a word X at step t+1 in the sequence is independent

of all previous words given that you have the word at step t. Clearly this is an erroneous

assumption to make about natural language but it turns out to be a quite useful one to make.

The n-gram allows us to store some history even given this independence by only looking back

n words. For example a bigram model would token the example sentence in the following way.

{“and  drank”,  “drank  coffee”,  “coffee  and”,  “and  talked”,  “talked  for”,  “for  an”,  “an  hour”}

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It is worth noting that in our case the terms are the same as the tokens for this sentence once

we switch to the bigram model.

For our purposes a unigram model will be used exclusively. As can be seen in our

example text, as n increases so does the number of unique terms. In our case our texts are

short (just a line) and our corpora are likewise small enough to not gain significantly from using

any more than a unigram model.

2.1.1 Representation of text

Now that we have established a method for taking our text and breaking it into

components we need to come up with a model for our text and ideally a vector space model. A

vector space model allows us to employ a wide range of standard mathematical treatments to

compare, understand, and explore a collection of text [28].

The simplest representation is fairly obvious: we have a collection of texts, we’ve  already  

disregarded word order, so we can simply say each text in the corpus is the collection of words

that compose that text. This is referred to as the bag of words model [28]. We can transform this

into a vector representation simply by using the entire set of terms as features, and then using a

binary indicator as to whether or not a term is present. This results in an individual text being

represented by a vector whose length is equal to the total size of the number of terms in the

document.

While this model is quite simple, in practice it turns out to be very useful. However there

is additional information we can add to our vector representation. The most obvious addition is

to convert our binary values into word counts. This results in the corpus being represented by a

matrix corresponding to the frequency count of terms in each document, referred to as the term

frequency matrix [28].

We can add an additional calculation to our model to allow it to contain even more

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information. Rather than simply looking at how frequently a term appears in a specific document

we  can  also  consider  how  frequently   it  appears  in  everything  else.  For  example  the  word  ‘the’  

may appear 3 times in one text, but we know that this word will appear all over the corpus in

great   frequency.  However   if  we  see   the  word   ‘duplicitous’  appear  3   times,  and  we   find   that   it  

appears dramatically less frequently throughout the rest of the corpus then there might be

something interesting it tells us about this specific text. To factor this in we can use the inverse

document frequency (i.e. 1/total count of terms in the corpus). This allows us to quantify how

special a term is. Finally we can factor this into our frequency count which allows us to arrive at

our final vector space model, the term frequency - inverse document frequency matrix (tf-idf)

[17, 27, 28].

Our final vector space model, the tf-idf matrix, has some interesting properties. First it is

a high dimensional matrix, with the number of features equaling the number of unigrams (unique

words) in our text. The size of the feature space can easily be tens or even hundreds of

thousands of features. Additionally most of the values in our tf-idf matrix are going to be 0, since

most text only contain a small fraction of the total vocabulary. This is referred to as a sparse

matrix [28].

2.1.2 Determining Similarity Between Texts

The core of our project is comparing how similar texts are. Since we now have a vector

representation of our text we have available to us all of the distance metrics we have for

measuring any two vectors. An initial assumption would be to compare the distance in

Euclidean space. That is, we can treat each vector as a point in n-dimensional space (where n

is the size of the vocabulary) and simply calculate similarity as the distance between these two

points, p and q as represented by the following equation [59]:

7

d(q, p)  =   (p  −  q ) +  (p − q )  +  . . . +  (p − q )  

Unfortunately, due both to the sparsity and high-dimensionality of our matrix, Euclidean distance

gives us less desirable results [28]. But we need not give up mathematical simplicity in order to

arrive at our similarity metric. If we think of our vectors not as points but as hyper-planes we can

then look at the angle between them. To make this even cleaner we can simply take the cosine

between the two vectors and get a nice value between 0 and 1 which will indicate how similar

the documents are [28].

sim (d , d )  =d ⋅ d|d ||d |  

2.2 Fractals

In his famous paper How long is the coast of Britain? Benoit Mandelbrot observed an

interesting fact about the coast of Britain [26]. If one is trying to measure the coastline, and

starts with a stick that is 200km long (Figure 2.1), you will find a length which would seem to

approximate the actual length of the coast, of course missing all the features that cannot be

measured by a 200km stick.

Figure 2.1: Measuring coast of Britain with a 200km measuring stick [55]

8

As one would expect if the length of the measuring stick is shrunk (as can been seen in Figure

2.2), a more accurate and also longer length will be arrived at.

Figure 2.2: Measuring coast of Britain with a 50km measuring stick [54]

Intuition tells us that we should expect that as you continue to shrink the measuring stick, you

would   eventually   converge   on   the   truth   length   of   the   coast.   However   Mandelbrot’s   brilliant  

observation was that this is in fact not the case. Instead it turns out that as you shrink the size

of the measuring stick the length of the coast continues to grow without limit!

The reasoning for this is rather interesting. As you zoom in on the coastline (within the

bounds of physical limitation of course) the shape of the coastline remains statistically self-

similar. That is the coastline as viewed from space will look similar to the shape if you were to

view it from a nearby hotel, and still the same if you were to walk up to the beach and look down

at your feet. Mandelbrot coined the term Fractal to label this scale invariant self-similarity.

A similar example is Brownian motion. In the following image (Figure 2.3) we show

Brownian motion over 100, 1000, 10,000 and 1,000,000 samples remains statistically self-

similar [26].

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Figure 2.3: Brownian motion at 100, 1,000, 10,000 and 1,000,000 samples

We can view the 100 samples as 1,000,000 samples zoomed in 10,000 times, and from these

images it is quite clear that there is no obvious difference, other than scale, in the shape of the

line. Just as the coast of Britain has the interesting property of being effectively of infinite length,

Brownian motion has the property of being everywhere continuous and nowhere differentiable

[26].

There are many patterns in mathematics that exhibit more obvious self-similarity, and

likewise strange properties [14, 40]. Take for example the Cantor set (Figure 2.4), which is

created by removing the middle third of a line, and then repeating this process with the two lines

created by removing this segment. Seven iterations of this process are visualized below [40,

56].

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Figure 2.4: Cantor set [57]

This fractal behavior of the Cantor set leaves it with the interesting property of having a measure

of 0 while also being uncountably infinite [40, 56].

Another interesting geometric fractal is the Sierpinski gasket (Figure 2.5), which is

created by continually removing the middle triangle from a large triangle as seen below [14, 40,

68].

Figure 2.5: Sierpinski gasket [69]

The Sierpinski gasket is a great example of a fractal that can be embedded in a 2 dimensional

space.

2.2.1 Fractals in Nature

As the coastline of Britain suggests, fractals are abundant in nature. They appear

everywhere from vegetables (Figure 2.6), to music, to the structure of toddler scribbles (Figure

2.7) and the distribution of galaxies in the universe [14, 24, 26, 40, 62].

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Figure 2.6: Romanesco cauliflower [58]

Figure 2.7: Doodles by Archer Kurt (age 3) compared to Levy flight

An obvious question that may arise when viewing fractals in nature is, “how   exactly   do   we  

define  a  fractal?”  While  there  is  no  absolute definition of a fractal, the key features we look for

are self-similarity and scale invariance [14]. Other features that can be useful are: that it is too

irregular to be described in traditional geometric language, it has a fine structure, and it has a

fractal dimension greater than its topological dimension [14].

Since our work is dealing with visualizing song lyrics, the fractal properties are heavily

12

constrained by the very discrete and finite nature of our similarity matrices. Nonetheless we will

still be able to identify a variety of the features that strongly suggest a fractal nature in the

structure of certain songs and poems.

2.2.2 Measuring Fractals

Given that fractals have such curious properties as 0 measure yet uncountably infinite,

or a coastline that has unbounded growth as the instrument measuring it shrinks, an important

question to answer is how are they measured? If we go back to thinking about a measuring stick

we can start to approach an answer. Suppose we have N measuring sticks of the length r and

we’re  trying  to  measure  the  circumference  of  a  circle.  We  will  have  the  following  formula  for  the  

approximate length (circumference) L of the circle.

L   =  N   ⋅  r

And for a normal geometric shape like a circle, as r approaches 0 we arrive at a finite limit which

is the circumference of our circle.

lim  →

N ⋅ r   =  L

The initial steps of this process are illustrated in Figure 2.8.

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Figure 2.8: Approximating circumference of circle

But as discussed previously, the fact that for fractals N ⋅ r never converges is precisely what

makes them interesting. However it turns out that the growth of fractals can be described as a

power law of the form N ⋅ r where the rate at which the length grows is constant with respect

to D [14,24,26,40]. This exponent is referred to as the Hausdorff dimension and is the typical

way to measure a fractal. We can alternatively view the previous formula as [40]:

D  :=   lim→

log  Nlog(1/r)

Beyond simply being a way to measure fractal, the Hausdorff dimension tends to intuitively

correspond to the weird dimensionality of fractals [40]. For example the Sierpinski gasket

mentioned previously, which is a 2 dimensional set that has 0 measure and is uncountable has

a Hausdorff dimension of 1.58. This seems to make intuitive sense as the Sierpinski gasket is

not   quite   a   2   dimensional   object,   owing   to   its   strange   ‘emptiness’   and   at   the   same   time is

certainly not simply a 1 dimensional line. Similarly the Cantor set has a Hausdorff dimension of

only 0.6309, once again nicely representing its nature of being less than a standard 1

dimensional line.

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The Hausdorff dimension is also useful for comparing fractals. For example the

coastlines of Ireland, Great Britain, and Norway (Figure 2.9) have Hausdorff dimensions of 1.22,

1.26, and 1.52 respectively [26, 62]. This gives a good sense of how jagged and deep the

respective coastlines are.

Figure 2.9: Coastline of Norway [66]

2.2.3 Measuring Fractals In Nature

Fractals such as the Sierpinski Gasket and the Cantor Set, despite their interesting

properties, due to their nature as mathematical sets have easy to determine values for N and r.

For purely mathematical fractals such as this, determining the Hausdorff dimension is very

straightforward; but what about fractals occurring in nature? For example the coastlines we

discuss, while certainly self-similar do not appear to possess easily identifiable values for N and

r. Clearly we are going to need alternatives to the Hausdorff dimension if we are going to assign

a fractal dimension to these objects.

The method used in calculating the fractal dimension of coastlines is referred to as the

box counting dimension [14, 40, 65]. This is defined as:

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D :=   lim→

log  N(ε)log(1/ε)

This is nearly identical to the Hausdorff dimension, except this time we have replaced r  with ε

where ε  is the length of the side of a box. The method involves placing boxes with progressively

smaller sides (defined by ε) around the coast. As ε  shrinks we continue to count the number of

boxes required to cover the coastline, which is N(ε). Eventually the ratio of the two logs will

converge to the box counting dimension D . A visualization of this can be seen in Figure 2.10.

Figure 2.10: Estimating the box counting dimension of the coast of Britain [60]

The box counting dimension works great for fractals that are similar to coastlines, which can be

represented as essentially a line; but what about fractals that are not easily represented by a

similar model? For example both goose down (Figure 2.11) and the Lichtenberg Figure (Figure

2.12) are fractal in nature [29], however cannot be easily measured by either the Hausdorff

dimension or the box counting dimension [40].

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Figure 2.11: Goose down [44]

Figure 2.12: 3-D Lichtenberg figure in acrylic [67]

For these types of naturally occurring fractals we use the mass dimension [40]. This method

works by expanding a circle of a continually growing radius R and observing a number of

particles that are contained in the radius, which describes the mass M. Thus mass can be

described by the following formula:

M   ∼ R

This translates into a very simple calculation for the mass dimension:

D   ∼  log(M)log(R)

17

This then allows us to empirically calculate the fractal dimension of naturally occurring fractals

that are not easily measure by either the Hausdorff dimension or the box counting dimension.

The mass dimension will be an essential tool in exploring the dimension of the fractals that

emerge in lyrical verse.

When looking at the various methods of calculating the dimension of a fractal, the key

observation  is  that  we’re  always  trying  to  understand  the  power  law  that  governs  the  self-similar

growth of the fractal structure.

2.3 Visualizing Text

The most prominent work in text visualization in general comes out of the work of

Fernanda Viégas and Martin Wattenberg. Well known for their general work in the field of

information visualization with their Many Eyes [47] project as well as more recent work in

visualizing wind patterns [49], the team has also done popular work within the general domain of

text visualization. Wordle, a tool for generating “tag   clouds” [5, 48] has become a popular

method of visualizing text. Figure 2.13 is the Wordle tag cloud for the background section of this

chapter. As can plainly be seen the visualization works by highlighting key words in the text

document  and  arranging  all  of  the  generally  important  words  into  a  ‘cloud’  shape.

18

Figure 2.13: Word cloud for the background section of this chapter

Despite the popularity of this visualization, it provides little additional information than simply

showing key words in a document.

Another text visualization coming out of the work of Viégas and Wattenberg is the

“Phrase  Net”  [46] which allows for the visualization of the way different words in a text relate to

one-another using a directed graph representation. Figure 2.14 is a visualization for the text of

Pride and Prejudice using this technique.

19

Figure 2.14: Phrase net  for  “Pride  and  Prejudice”  [36]

Viégas  and  Wattenberg  also  produced  a  similar  graph  representation  in  their  ‘Word  Tree’  [50].

Rather than showing relationships between words the Word Tree acts as a visual concordance,

allowing those interacting with the visualization to observe all possible next words in a sentence

from a given word. An example taken from Martin Luther King’s  “I  have  a  dream  speech”  is  seen  

in Figure 2.15.

20

Figure 2.15: Word  tree  for  “I  Have  a  Dream”  [70]

2.3.1 Other Examples of Text Visualization

The Storylines project [79] attempts to use latent semantic analysis (LSA) to allow users

of the visualization to explore themes in unstructured text (Figure 2.16). LSA works by using the

Singular Value Decomposition (SVD) to project the very high dimensions tf-idf matrix onto a

lower dimensional subspace. One of the byproducts of this process is that features with similar

meaning will be projected near each other, and as a result a certain degree of semantics can be

inferred [28]. Once SVD is performed on the tf-idf matrix then terms are plotted in clusters to

create a semantic network. The aim of this visualization is that clusters of meanings will be

visible in the semantic network providing insight into themes present in the document.

21

Figure 2.16: Image of latent semantic similarity [79]

The Paper Machines project [32], for the Zotero citation and research platform, combines many

of the previously mentioned visualization in order to help its users understand and explore texts

they are currently using for research. The two primary visualizations it employs are word clouds

and phrase nets.

S. Posavec has combined art and information visualization in her work exploring the

structural   nature   of   text   and   its   aesthetic   properties.   In   her   piece   “Writing   Without   Words”  

(Figure 2.17) [38] Posavec represents and compares the writing styles of a variety of authors by

focusing strictly on the properties of the text including: sentence length, parts-of-speech,

sentence rhythm, and punctuation.

22

Figure 2.17:  “Writing  Without  Words”  [76]

Her work contains many other examples of similar explorations of the aesthetic nature of the

structure of text itself [37].

2.3.2 Visualizing Poetry and Song Lyrics

The work our project focuses on is strictly concerned with visualizing repetition in lyrical

verse. There are a variety of text visualization projects that concern themselves specifically with

poetry and song lyrics. Abdul-Rahman et al. examined a rules based system for visualizing

poetry [1]. Their project worked by visually representing a variety of features of the poem:

phonetic relations, phonetic features, words, and semantic relations. The result of their

technique on a sonnet can be seen in Figure 2.18

23

Figure 2.18: Visualizing a sonnet [1]

This work was then built into the Poem Viewer at the University of Oxfords e-Research Centre

(http://ovii.oerc.ox.ac.uk/PoemVis/index.html). The Poem Viewer allows users to upload poetry

and then interactively visualize the various components of the poem. Figure 2.19 gives a

screenshot of this interface.

Figure 2.19: Screenshot from POEM Viewer

Another researcher, Rhody, used the MALLET tool [25] in order to aid in the

understanding of specific collection of poems [39]. The project consists of using the MALLET

tools to perform topic modeling in order to identify potential topics in the corpus of 276 poems.

Rhody then utilized a series of standard charts and graphs to better understand the topic

24

modeling, and gain insight into themes present across poems in the corpus.

In 2005 the Group Lab at the University of Calgary did work in visualizing song lyrics as

they occur in a song in real-time [10]. The visualization works by highlighting each word as it is

sung, and then visually representing how long the word is vocalized by the singer with a growing

radius representing the duration that the word has been sung. Figure 2.20 shows this in action.

Figure 2.20:  Screenshot  of  Group  Lab’s  lyric  visualization

Another song lyric visualization project by J. Oh [30] seeks to improve the way that

musicians and performers can understand how song lyrics are incorporated into the larger work

of   the   song.   Oh’s   project   tracks   linguistic   features,   musical   features,   and   the   overall   song  

structure in order to aid the reader in understanding where and how specific sections of the

lyrics fit in with the rest of the song. Accented syllables are italicized, key words are bolded, the

pitch is represented as a step chart, and various other musical components are encoded into

25

the visualization (Figure 2.21).

Figure 2.21: J.Oh lyrics visualization [30]

One of the most interesting works in visualizing the lyrics comes from S. Posavec and G.

McInerny to create the artwork in a booklet included in the album, Of The Blue Colour of The

Sky, from the popular music group OK Go [6]. The booklet contains many representations of the

lyrical text visualized in different ways. One compares the syllables in the songs (Figure 2.22)

with the syllables in another book by A. Pleasonton, The Influence of the Blue Ray of the

Sunlight and of the Blue Colour of the Sky, an early work inspiring the field of chromotherapy

which influenced the name of the album [52].

26

Figure 2.22: OK Go booklet – syllable visualization [31]

Another visualization represents the various words in the lyrics and their parts of speech which

are represented by color (Figure 2.23).

Figure 2.23: OK Go booklet – parts of speech visualization [31]

27

While the aim of these visualizations is to create an aesthetic experience rather than to

necessarily provide information about the lyrics, this work demonstrates quite well that the

patterns of data in lyrical verse contain within themselves aesthetics properties.

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Chapter 3

Visualizing the Structure of Repetition in Lyrical Verse

“You  say  I  am  repeating  something  I  have  said  before,  shall  I say  it  again?”  

-- T.S. Eliot, The Four Quartets

3.1 Overview

When analyzing lyrical works, whether poetry or songs, repetition plays a major part in

understanding the work. Repetition can manifest itself in many ways: rhyming verses, recurrent

themes, structured choruses, allusions to earlier works, attention to select words, and meter to

name a few. Typically, in the close reading of lyrical works there is no systematic way to

observe patterns of repetition and the reader is left to discover them only through the process of

careful note taking and continual rereading of works.

After many years of rereading T.S. Eliot’s  “The  Four  Quartets”,  and  copious  notes  in  the  

margins pointing to other parts in the stanza or even the large poem, it became clear that there

must be a faster, more holistic way to view repetition in the work as an entirety. Perhaps even to

form a framework for tracking similarities in the structure of repetition across multiple works and

authors.

3.2 Visualizing Repetition

For the most part, the intersections between those who study the content of lyrical verse

and those who study computational models of text are small. As such there is a lack of attempts

to visualize this important feature of text. By far the most dominant methods of visualizing text

are to either represent it in a graph structure [1, 32, 38, 46, 50, 79] or to provide line level

annotations [1, 10, 30]. While the graph-based visualization allows for the visualization of much

29

of the connecting structure of text it is usually at the cost of losing sequential information about

this structure. Themes and some forms of repetition can be observed, but the actual structure of

repetition is lost. With the annotated line approach sequential information is preserved but at the

cost of losing a  “big  picture”  view of the larger structure of the work. Attempts to blend these two

[1] end up with extremely dense representations that can be difficult to parse and understand

when trying to view repetition as a whole. The challenge our visualization tackles is concisely

capturing the structure of repetition within an individual work, so that this structure itself can be

easily studied. The primary aim of our visualization is to gain insight into how repetition is

constructed in lyrical verse.

While there are many forms of repetition in lyrical verse, a good place to start is focusing

on repetition that is already easy to deal with given the existing tools of Natural Language

Processing. For our present work our focus is the exploration of repetition that can be

expressed by determining the similarity in content using the cosine between vectorized

representations of text. By simply visualizing a matrix of cosine similarity using a color gradient

for values from 0 to 1, we are able to observe a surprising number of previously difficult to

observe patterns of repetition in lyrical works.

3.3 Why Visualize?

While we have hinted at some of the potential benefits of visualizing lyrical work, our

visualization technique elucidates two major areas of understanding the individual works:

qualitatively by adding an extra layer of insight into the work which can aid in the process of

close reading, and quantitatively by providing tools by which the lyrical work can be understood

through mathematical modeling.

3.4 Close Reading

Close reading [3] is essentially the approach to understanding literature which involves

30

focused study of the specific portions of the work, the language it uses, and its relation to the

overall structure of the work as a whole. Due to the focused nature of close reading it can be

particularly hard to observe the work as a whole. However this, somewhat paradoxically, leads

to the problem that the way each component relates to the whole can be difficult to see despite

being potentially very insightful. By visualizing the entire work in a relatively compact space our

visualization allows for the structure of the poem itself to be subject to close reading, something

that is otherwise difficult if not impossible to achieve. In Figure 3.1 and Figure 3.2 we see our

visualization for William  Blake’s  “Divine   Image”   [2] alongside the original text. Despite being a

short poem it can be easily seen that the structure of repetition in the poem is not inherently

obvious until it is visualized. As an example it is clearly seen from our visualization that the first

line is echoed throughout the poem, though rarely is it exactly reproduced.

Figure 3.1: Line similarity for William Blake’s  “Divine  Image”

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Figure 3.2: Songs of Innocence - Divine Image [61]

In this way our visualization can serve as a valuable aid to understanding the stylistic use of

repetition as well as finding themes that may emerge in the work based around the use of

similar language.

3.5 Mathematics of Aesthetics

From a quantitative perspective this visualization allows viewers insight into something

novel, rather than merely aiding in existing analysis. By not simply observing the pattern of

repetition in an individual poem or song we can begin to, at a glance, observe patterns that

appear across multiple poems or songs. Since we are essentially visualizing a matrix of

numerically represented similarity, this then opens up lyrical verse to a wide range of

mathematical analysis.

While repetition in lyrical verse is often an essential component (especially in forms such

as popular music) an interesting paradox emerges when we start to think on the meta-level

about how the structure of repetition itself works. Patterns are by their nature predictable, but

repetition that is too predictable is boring. At the other extreme, repetition that is completely

32

unpredictable, i.e. without pattern, is immediately shut out by the listener. On the one end of

this spectrum we have a record perpetually skipping and the other we have pink noise. How can

we create a tradeoff between these two extremes? Certainly a simple compromise between the

two seems deeply flawed. As a mental exercise imagine a record of pink noise continuously

skipping. Such an experience would certainly be even less appealing aesthetically than either of

its components!

If we wish to model aesthetics we need to find a model of the world that simultaneously

allows for the emersion of complex patterns while remaining, ultimately, unpredictable.

Fortunately there is already a field of study which focuses explicitly on this phenomenon. In the

study of dynamical systems, i.e. Chaos Theory, we come across systems that while entirely

deterministic, vary extremely from initial conditions and whose development cannot be predicted

[40]. And of course the most familiar manifestation of this is the fractal.

Perhaps the greatest insight gained by visualizing repetition in lyrics is the clear

emergence of self-similarity in the structure of repetition within works containing very dense

structures of repetition. Looking at classic Dr. Seuss poems as well as a variety of popular

music we can not only begin to observe this fractal nature in lyrical verse but actually measure

it.

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Chapter 4

Implementation

“It’s  you,  it’s  you,  it’s  all  for  you Everything I do I tell  you  all  the  time” -- Lana Del Rey, Videogames

4.1 Preprocessing

Before any work can be done, first it is necessary to get the raw text data in a form that

can be used by a program. This process begins with manual annotation of the text. The most

fundamental preprocessing step is to manually remove all content from a text document which

is not directly related to a poem or song. For example: many collections of poetry in Project

Gutenberg contain introductions to poems or sections; all of these need to be removed before

the text can be processed.

Since we wish to be able to work with any sort of lyrical verse we need a methodology

that would generally be able to separate a body of work (e.g. in the case of music and album, or

in the case of poetry the larger poem itself or book containing smaller poems) from its

subdivisions (e.g. an individual song or a stanza). We want to extract each individual line from a

source of lyrical verse and annotate it with the work, and subdivision name (e.g. album-song-

verse, poem-stanza-line)

Another constraint is that potentially large amounts of plain text documents would need

to be hand annotated, this makes conventional markup choices, such as xml, suboptimal due to

the amount of time required to hand format the entire document. Since most text is coming from

34

sources like Project Gutenberg, in which titles and appropriate line breaks could be assumed a

simple way  of  marking  up   text  was  devised.  Text  surrounded  by   ‘==’  would   set the work title,

and text surrounded  by   ‘__’  would  set   the  subdivision   title.  Pseudo code for the algorithm for

consuming text is as follows:

while readline(): if  line  matches  “==(.+)==”: set  work_title  to  the  value  in  between  the  ‘==’

else  if  line  matches  “__(.+)__”: set subdiv_title  to  the  valu  between  “__” else if line is a character string: write work_title, subdiv_title, line to file else line contains nothing, ignore.

The text parsed in this manner is then outputted to a .csv file. A simple csv file is chosen

because many tools easily support this file type.

4.2 NLP

4.2.1 NLP Preprocessing

Once we have the data in a machine readable csv file we can now begin the process of

NLP work. We have done this in R using  the  packages  ‘tm’ and ’lsa’  [15,16,71]. When we read

in the csv file output by our preprocessing steps the result is 3 columns of data: work name,

subdivision name, and the text for the line itself. Before we can do any work we need to perform

some basic clean up. All excess white space and all punctuation is removed and all strings are

transformed to lower case.

4.2.2 Common NLP Preprocessing Avoided

It is fairly common in Natural Language Processing to remove what are referred to as

35

stopwords from a text. Stopwords are   common  works   such  as   ‘as’,   ‘the’,   and   ‘and’.   Typically  

these words act simply as noise, detracting from words which give more information about the

text. Another common NLP preprocessing technique is stemming which attempts to

automatically truncate words. This would cause words such as ‘walked’   and   ‘walking’   to   be  

reduced  to  the  same  root  ‘walk’. While both of these steps are fairly common in any nlp pipeline

they are deliberately avoided in this case due to the density and deliberate nature by which

lyrical verse is constructed.

4.2.3 Vectorization

The key step is to create a vectorized representation of the text. To do this we need to

create a term-frequency inverse-document frequency (tf-idf) matrix. This representation allows

us to capture a great deal of information about our text data. Rather than simply calculating

either the Boolean presence of a word in a vector, or its raw frequency count, we can gather

more information into our vector by also taking into account the terms frequency times the

inverse of the number of times that term appears in other documents. This way the uniqueness

of a term in the corpus can also be represented. To  automate   this   process  we  used   the   ‘tm’  

package in R.

poem.dtm <- DocumentTermMatrix(poem.corpus,control = list(weighting =

function(x) weightTfIdf(x, normalize = FALSE), wordLengths=c(1,Inf)))

One   important   note   on   the   code   here   is   that   the   default   behavior   of   the   ‘tm’ package is to

remove words that are less than 2 characters. In the majority of NLP tasks this would be a

perfectly sensible decision. However, as we will see in the results sections, it is not uncommon

for a two character, or even a single character, word to have a large impact on the language of

the lyrical work. This is a recurring issue with NLP involving poetry, minutiae which is often

36

noise in large text documents is actually very useful information in poetry. The majority of text

preprocessing that is standard in most text vectorization processes is in fact detrimental when

working with a corpus in which each word, no matter how small, was likely chosen with

intention.

4.2.4 Sparse Representation

One of the challenges in working with text is the high dimensionality of the feature space

(which is the complete vocabulary of the corpus) of the vectorized representation. Because the

vast majority of words in the total feature space are not present in an individual line (which

frequently is only 5-6 words) we are left with an incredibly sparse matrix. To more efficiently

work with text data we keep our matrix represented in a sparse form, where each value is stored

at an index and all others are assumed to be 0. While this leads to a dramatically more efficient

representation there are often functions which require us to transform our data not only from

sparse matrix to full matrix, but also between different sparse representations based on what

different R packages expect data to look like.

4.3 Visualization

Now that we have our data in a vectorized form we need to find a way to compare lines

(as well as works). Cosine similarity is the standard for information retrieval and natural

language processing. What we want is to view the similarity between every line to every other

line. So for an N x M matrix where N is the number of lines in the poem or song and m is the

size of the vocabulary we take the cosine between every N

sim_matrix for i in N: for j in N: sim_matrix[i,j] = cos(i,j)

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Conveniently  the  R  package  ‘lsa’  [71]  contains  a  function  ‘cosine’  which  will  compute the cosine

between all column vectors. Because we want to compute the cosine between rows we simply

need to transpose the matrix and then transpose it back again:

sim.m <- t(cosine(t(as.matrix(poem.sm))))

Now we have an NxN matrix representing the similarity between every line with every other line.

This of course means that all items along the diagonal of this matrix will be perfectly similar

since   this   represents   each   line’s   similarly   with   itself. There is also a mirroring effect in the

visualization since cosine(x,y) = cosine(y,x). While this does create redundant data in the

visualization, this redundancy does not necessarily detract from the ability to interpret the data

and, by providing an alternate perspective, may in fact enhance it.

One of the more interesting challenges was with orientation. In the first draft the lower

left corner was 0,0 as is typical for a Cartesian graph. However in showing this information to

public  audiences  one  of  the  most  common  critiques  was  “Why  are  they  backwards”? Although

using a standard Cartesian plane obviously made sense, we decided in the end to have the 0,0

coordinate located in the upper left corner. There are 2 major reasons for making this decision.

First, this is somewhat standard for the already popular correlation matrix, and most importantly

English speakers read from left to right, and top to bottom. Especially when juxtaposed with the

original poem this second method of visualizing the work seems radically superior.

Finally we need to actually create our visualization. Thanks to the powerful R

visualization   library   ‘ggplot2’   [51], this was relatively simple given that we had already created

our similarity  matrix.  Using  the  package’s  qplot function and simply specifying the geometry to

be  ‘tile’  our  similarity matrix is automatically converted into an image such as Figure 4.1. Only a

few minor tweaks of graphical parameters were required to clean up the overall display.

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Figure 4.1: Line similarity for  Vampire  Weekend’s  “Ya  Hey”  [23]

4.4 Computing Fractal Dimension

Since our representation of text is necessarily discrete and only encompasses a rather

small scale it was essential to find a method of calculating the fractal dimension that would still

perform well given these constraints. For this we chose to use the mass dimension (details in

Chapters 2 & 5) which attempts to infer fractal dimension by estimating the power relationship

between a radius r and the mass M representing the mass of points contained within an

increasing radius.

One of the key implementation differences between representing similarity in this

calculation was the necessity of choosing a threshold for ‘similarity’  which  would  decide which

points were included in the mass calculation. In our visualization the entire spectrum from 0 to 1

could be represented, however for the sake of calculating mass we stuck with threshold of 0.5

based simply because it seemed to most honestly represent the similarity observed. Once a

threshold is set then the distance between each point is calculated from an origin (almost

always the last line in the song) which visually appeared to be the origin of the fractal.

distances.of.points <- sapply(1:length(points.x),function(i){ sqrt((points.x[i]-o.x)^2 + (points.y[i]-o.y)^2) })

39

Then for a discrete range of radii, the algorithm simply collects how many points are closer to

the center than r (i.e. which points are contained in the radius).

point.count <- sapply(radii,function(r){ sum(distances.of.points < r) })

Now we have a collection of points the sequence of radii as well as corresponding mass for

each. We then used linear regression to find the approximate slope of these lines. To do this we

used  R’s  glm  function  which  creates  a  generalized  linear  model  for  the   log  of  the  masses  and  

radii. Taking the log of each is necessary since the exponential growth can then be represented

as linear growth.

Rather than using ordinary least squares (OLS), we used iteratively reweighted least

squares (IRWLS). IRWLS unlike OLS does not assume a consistent variance between the x

and y values. This ends up weighing the denser cluster of values more strongly than the initial

set. This is useful for two reasons. First, due to the discrete nature of the values it is visibly clear

that the initial points have much higher variance; second, due to the log transformation the

points at the end of the sequence are more densely clustered.

4.4.1 Visualizing Fractal Dimension

The visualization for the mass dimension calculation consists of two parts. The first,

show in Figure 4.2, visualizes the computation process showing how progressively large radii

enclose increasing more of the mass.

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Figure 4.2: Calculating the mass dimension

The second visualization, shown in Figure 4.3, is simply the log-log plot of M to r with the linear

regression line showing the slope between the two points (which is the approximate fractal

dimension).

Figure 4.3: Log-log plot for approximating mass dimension

Both of these plots were created with R’s built-in plotting functionality.

41

Chapter 5 Results

“Karma  police,  arrest  this  man He talks in maths He buzzes like a fridge He's  like  a  detuned  radio” -- Radiohead, Karma Police

5.1 Visualization - Simple Applications

The original motivation for this visualization was the observation that when reading

larger poems it became exceedingly difficult to track themes and repeated lines throughout the

entire work. Close reading of a text is typically focused on only a small section of the work at a

time. Understanding the relationship a given section has to another can be tremendously

important to understanding the work as a whole and even special significance an individual

subsection may have because of the role it plays with other subsections. As more of these

visualizations were created it became apparent that this was also an issue even in smaller,

denser works. In the following sections, we explore how close reading can be aided by viewing

the structure of repetition. We also explore how one can focus strictly on the structure of lyrics in

a song, and essentially use the lyrics to understand the structure, inverting the process in the

first example.

5.1.1 Visualizing the Thunder - Visualization and Close reading

Without the aid of visualization it is very hard to arrive at a holistic view of meaningful

patterns of repetition in a work. One of the most valuable functions of these visualizations is

they allow expansion upon traditional close reading. For example, let us look at the final section

42

of  T.S.  Eliot’s  The Wasteland [12]. The section is titled  “What  the  Thunder  Said”.    Figure 5.1 is

the line similarity visualization for this section.

Figure 5.1: Line  similarity  for  T.S  Eliot’s  “The  Wasteland  - what  the  thunder  said”

Here we see the common pattern of a single primary cluster of repetition, in this case at the

beginning of the section. Taking a look at the early parts of the poem we can see where in the

text some of this repetition is occurring:

If there were water And no rock If there were rock And also water And water

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A spring A pool among the rock If there were the sound of water only Not the cicada And dry grass singing But sound of water over a rock Where the hermit-thrush sings in the pine trees Drip drop drip drop drop drop drop But there is no water

The  words  which  are  causing  most  of  the  repetition  are  ‘water’  and  ‘rock’.  Particularly  interesting  

is  that  this  passage,  ending  with  the  lament  “But  there  is  no  water”,  is  also  the  end  of  repetition.  

It is not much of a stretch  to  point  out  that  the  repetition  of  words  replicates  the  “pitter  patter”  of  

water on rocks, of rain drops. This can literally be seen in the text when visualized. Something

that is not obvious without the visualization.

When looking for further repetition something truly fascinating sticks out. Near the end of

the passage the repetition of language starts to return, most clearly evident in the 6 solid dots

appearing in the last section (Figure 5.2).

Figure 5.2: Close-up  of  “DA”  section

These 3 lines (6 dots due to the mirroring effect) are  actually  one  single  word  repeated:  ‘DA’.  To  

truly see what is interesting it is important to understand their context:

Then spoke the thunder DA

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The sound  ‘DA’  which  is  the  focus  of  the  repetition  in  this  section  is  the  voice  of  the  thunder,  the  

sound of a coming storm. Looking back at the overall picture we see the transition from imagery

of water, to its absence, to its eventual return. All of this is in fact visible simply from the pattern

of repeated words. We can observe the cloud dissipate and reemerge with the sound of

thunder.

5.1.2 Understanding the Structure of Popular Songs

While poetry has relied heavily on repetition and recurring themes, the tendency towards

repetition in song lyrics is even stronger. Not only do choruses and refrains create segments of

perfect repetition, it is not uncommon to find a series of phrases that continue being slightly

altered in progression. We can see this in the example of Vampire  Weekend’s  song   ‘Walcott’,

illustrated in Figure 5.3 [22].

Figure 5.3:  Line  similarity  for  Vampire  Weekend’s  “Walcott”

45

Here there is a clear pattern of phrases that repeat in a similar but not exact fashion.

If we look at the opening lines we can inspect what is happening here:

Walcott, don't you know that it's insane? Don't you want to get out of Cape Cod, out of Cape Cod tonight? Walcott, Mystic seaport is that way Don't you know that your life would be lost out of Cape Cod tonight? Walcott, don't you know that it's insane? Don't you want to get out of Cape Cod, out of Cape Cod tonight?

As we can see there is a repeating pattern of phrasing in lines 1, 3, 5 and 2, 4, 6. This is what is

creating  the  ‘checkerboard’  effect.

What becomes more interesting is that with such stronger structure to repetition, we can

also start to see patterns among the song lyrics themselves. Most obvious when viewing the

entire  collection  of  songs,  for  example  every  song  on  Vampire  Weekend’s  Modern Vampires of

the City [45] in Figure 5.4, is that nearly every song has a large cluster of repetition.

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Figure  5.4:  Line  similarity  for  all  songs  on  the  album  “Modern  Vampires  of  the  City”.  N.B.  The  song  “Young  Lion”  consists  only  of  1  line  repeated  4  times  hence  the  solid  block.

This is often just the chorus, but as we further examine these clusters we begin to realize that

there are patterns in these as well. For example many songs build up repetition as they

progress, exhibiting virtually no repetition in the earliest stages and increasing until the end.

Others invert this pattern. This is fascinating as we can see a similar pattern of structure

between  Radiohead’s  ‘Karma  Police’  [75] (Figure 5.5) and what we see in the sections of T.S.

Eliot’s  The Four Quartets, Burnt Norton III (Figure 5.6) and East Coker III (Figure 5.7)[11].

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Figure 5.5: Line  similarity  for  Radiohead’s  “Karma  Police”

Figure 5.6:  Line  similarity  for  “Burnt  Norton  III”

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Figure 5.7:  Line  similarity  for  “East  Coker  III”

At this point it should be clear that there are many observations about the structure of lyrical

representation that benefit from visualizing the cosine similarity of the documents. Even more

striking is the ease with which repetition can be compared across genres and authors. Seeing

similar patterns in repetition between Eliot and Vampire Weekend is something that would be

very difficult to achieve analyzing only the text itself.

5.2 The Fractal Nature of Lyrical Verse

Starting from the simple observation that there are recurring patterns of similarity across

a wide range of lyrical verse, we stumble across an even more startling observation. Among

these families of patterns we actually come across many that are truly fractal in nature. This is a

significant insight in that it provides a much deeper understanding of why and how the

mechanics of repetition are able to create aspects of the aesthetic experience.

5.2.1 Green Eggs and Ham and Self-Similarity

One of the most fascinating insights provided by visualizing lyrical verse is what appears

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when we start looking at either song lyrics with repeated chorus and verses or very repetitive

dense poetry such as is typically found in books of children's verse. The most striking of these is

Dr  Seuss’s  classic  “Green  Eggs  and  Ham” (Figure 5.8) [42]. For those unfamiliar with the work

one of the key structural components is the building up of an enumeration of conditions in which

the central character will not eat green eggs and ham:

I could not, would not, on a boat. I will not, will not, with a goat. I will not eat them in the rain. I will not eat them on a train. Not in the dark! Not in a tree! Not in a car! You let me be! I do not like them in a box. I do not like them with a fox. I will not eat them in a house. I do not like them with a mouse. I do not like them here or there. I do not like them ANYWHERE!

This list is creating a recurring pattern of repetition. Along with other mechanisms of repetition

within the larger poem this leads to a particularly striking visualization.

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Figure 5.8: Line similarity  for  “Green  Eggs  and  Ham”

Aside from the sheer complexity of the structure of repetition there is something else which is

very crucial to truly understanding the significance of this structure. The careful observer will

notice that the square pattern between lines 1 through approximately line 40 is self-similar to the

structure over the rest of the work (Figure 5.9).

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Figure 5.9 Zoomed  in  segment  of  “Green  Eggs  and  Ham”  (compare with entire document in

Figure 5.8)

Not only is the self-similarity present between the beginning and the body of the work but

something we see all over at various scales (highlighted in Figure 5.10). This self-similarity

across  scales  implies  that  the  structure  of  repetition  in  “Green  Eggs  and  Ham”  is  in  fact  fractal in

nature.

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Figure 5.10: Highlighted self-similarity  in  “Green  Eggs  and  Ham”

This is quite an astounding discovery and has broad implications for how the aesthetics

of   repetition   actually   work.   We’re   used   to   seeing   fractals imagined in a variety of natural

phenomenon from finance to cauliflower but without visualization it certainly not clear that the

mechanics  of  children’s  poetry  may  in fact be driven by a hidden fractal.

Taking   other   examples   from   Dr.   Seuss’s   body   of   work,   with   an   eye   open   for   self-

similarity, we come across more self-similarity, though not as obvious  as  in  of  “Green  Eggs  and  

Ham”. In   “Fox   in  Sox”   [41] there is a common pattern of a dense cluster followed by sparser

areas of repetition (Figure 5.11). This pattern is somewhat apparent in work as a whole but also

appears, sometimes with reversed orientation throughout the poem.

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Figure 5.11: Line  similarity  in  “Fox  in  Sox”  with  self-similarity highlighted

In  “Hop  on  Pop”  [43] we seem a similar structure (Figure 5.12) repeated at all scales (though not

present in the overall structure of the work).

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Figure 5.12:  Line  similarity  in  “Hop  on  Pop”  with  self-similarity highlighted

Not  every  Dr.  Seuss  poem  presents   itself  as  a   fractal  quite  as  clearly  as   “Green  Eggs  

and  Ham”; however we consistently find the presence of some degree of scale invariant self-

similarity. More than mere curiosity, this provides insight into the way fractals might work to

structure repetition in order to create an overarching aesthetic.

5.2.2 Searching for Fractals in Song Lyrics

The  reason  that  we  are  able  to  so  easily   identify  fractals  in  works  such  as  “Green  Eggs  

and   Ham”   is   that   the   density   of   the   repetition   is   more   extreme   than   a   work   such   as   “Four  

Quartets”.  In  order  to  continue  our  search  for  fractal patterns in lyrical verse, looking at lyrics in

popular music is likely to be another source rich in dense repetition.

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Most popular music contains lyrics that work on a common theme and make heavy use

of repeated chorus and very similar verses which echo each other. One of many examples of

this  echoing  effect  (with  both  exact  and  similar  repetition)  can  be  seen  in  Tame  Impala’s “Why  

Won’t  They  Talk  to  Me”  [35]

Out  of  this  Zone,  trying  to  see,  I’m  so  alone,  nothing  for  me I  guess  I’ll  go  home,  try  to  be  sane, try to pretend, none of it happened … Out  of  this  zone,  Now  that  I  see,  I  don’t  need  them  and  they  don’t  need  me I  guess  I’ll  go  home,  try  to  be  sane,  try  to  pretend,  none  of  it  happened

Given how common similar lyrical structure is to popular music we should expect to see much

stronger patterns in repetition (Figure 5.13).

Figure 5.13:  Line  similarity  for  Tame  Impala’s  “Why  Won’t  They  Talk  to  Me”

And  as  expected  line  similarity  in  the  visualization  of  “Why  Won’t  They  Talk  to  Me”  certainly  has  

a stronger visual repetition. However, it seems reasonably clear here, that what we are seeing

possesses no obvious self-similarity.

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We  don’t  have  to  look  too  far,  however  to  find  songs  that  do  exhibit more obvious self-

similarity. In  “Feels  Like  We  Only  Go  Backwards”  [48]  (Figure 5.14) what we see, noted by the

numerous bright spots, is that almost all of this self-similarity is created through the repetition of

the chorus which appears as follows:

chorus verse chorus verse chorus chorus chorus

When we look at the actual text of the chorus:

It feels like I only go backwards, lately Every part of me says go ahead I got my hopes up again, oh no, not again Feels like we only go backwards darling

We see that there is a sub-repetition in the chorus itself between the nearly identical (but not

perfectly identical) first and last lines.

Figure 5.14: Line  similarity  for  Tame  Impala’s  “Feels  Like  We  Only  Go  Backwards”

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Not only does the repetition form an interesting pattern but we once again see scale invariant

self-similarity. To see this clearly, look at the box surrounded by a border formed in the lower

left, then notice that this box itself is followed by a similar border.

Perhaps more interesting are cases in which inner structure of a verse is used to create

self-similarity  with  the  overall  structure  of  the  chorus.  In  “Feels  Like  We  Only  Go  Backwards”  the  

primary function of the verses in the pattern is to create essentially white space so that the

larger pattern created by the chorus can emerge.

In Vampire Weekend’s   “Obvious  Bicycle”   [20]  we find a more sophisticated pattern of

repetition (Figure 5.15), in which the first 10 lines mirror the similarity structure of the chorus.

Figure 5.15: Line  similarity  for  Vampire  Weekend’s  “Obvious  Bicycle”

The self-similarity is highlighted in Figure 5.16 for clarification

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Figure 5.16: Vampire  Weekend’s  “Obvious  Bicycle”  with  self-similarity highlighted

It   is  worth  noting  that  while  “Obvious  Bicycle”  and  “It  Feels  Like  We  Only  Go  Backwards”  both  

exhibit a clearly fractal structure with exemplified scale-invariance, the direction that invariance

progresses   through   the   song   is   different.   In   “Obvious   Bicycle”   as   the   song   progresses we

effectively  magnify   the   first  section,  whereas   in   the  “It  Feels  Like  We  Only  Go Backwards”  we  

see the opposite scaling effect  (that  of  “zooming  out”). This further adds to an understanding of

ways in which fractal behavior can generate different aesthetic properties.

Looking through an increasingly large collection of songs we continue to come across

obvious fractals. What become particularly interesting are the fractal patterns that begin to

emerge  across  various  songs.  Take  a  look  at  Radiohead’s  “Subterranean  Homesick  Alien”  [77]

from the album OK Computer (Figure 5.17).

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Figure 5.17: Line  similarity  for  Radiohead’s  “Subterranean Homesick  Alien”

And then compare that with another song from the  same  album  “The  Tourist”  [78] (Figure 5.18).

Figure 5.18: Line  similarity  for  Radiohead’s  “The  Tourist”

Here we see an instance of self-similarity which occurs not within a single song but across many

others.  Again  we  can  see  this  same  pattern  emerge  on  Radiohead’s  prior  album  The Bends in

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“Bullet  proof…  I  wish  I  was”  (Figure 5.19) [73].

Figure 5.19: Line  similarity  for  Radiohead’s  “Bullet  proof  ….  I  wish  I  was”

And again we see a similar pattern on the album  in  the  song  “High  and  Dry” (Figure 5.20) [74].

Figure 5.20: Line  similarity  for  Radiohead’s  “High  and  Dry”

Even   though   the   pattern   in   “High   and   Dry”   is   very   similar   to   the   other   3   examples   from  

Radiohead, we also get a glimpse of this same pattern starting to fall into a pattern of self-

similarity. The scale invariance is seen in the bordered rectangle in the right-left corner

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expanding in a similar fashion as its own border becomes the end of the square in the next

series of repetition.

Not only do we see repetition in the very structure of repetition within one artist’s works,

but across multiple music groups. Here we can see a very similar pattern emerging between

Vampire   Weekend’s   “Hudson”   (Figure 5.21) [19] and Lana Del   Rey’s   “Million   Dollar   Man”  

(Figure 5.22) [8].

Figure 5.21:  Line  similarity  for  Vampire  Weekend’s  “Hudson”

Figure 5.22: Line similarity for Lana  Del  Rey’s  “Million  Dollar  Man”

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And  again  between  “Feels  Like  We  Only  Go Backwards”  (Figure 5.23) [33]  and  Lana  Del  Rey’s  

“Radio” (Figure 5.24) [9]

Figure 5.23: Line  similarity  for  Lana  Del  Rey’s  “Radio”

Figure 5.24: Line  similarity  for  Tame  Impala’s  “Feels  Like  We  Only  Go  Backwards”

This repetition of fractals seems to imply that the repetition driving song lyrics follow a self-

similar pattern, but also across popular music in general we can expect to see a broader system

of self-similar representation. That is, a common set of fractals are able to create a broad range

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of lyrical verse.

5.2.3 Song Lyrics and Cantor Dust

Perhaps the most dominating repeating pattern of repetition is a series of squares, which

without a larger context do not seem to obviously invoke a fractal. A wide range of songs from

an equally wide range of artists demonstrate  this  ‘boxy’  repetition.  This pattern can be seen in

Lana  Del  Rey’s  “Diet  Mountain  Dew”  (Figure  5.25),  Radiohead’s  “Climb  Up  the  Walls”  (Figure  

5.26),  Vampire  Weekend’s  “one”  (Figure  5.27),  and  Radiohead’s  “Ripcord”  (Figure  5.28)  [7, 21

,72, 76].

Figure 5.25:  Line  similarity  for  Lana  Del  Rey’s  “Diet  Mountain  Dew”

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Figure 5.26:  Line  similarity  for  Radiohead’s  “Climb  Up  the  Walls”

Figure 5.27:  Line  similarity  for  Vampire  Weekend’s  “one”

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Figure 5.28: Line similarity for Radiohead’s  “Ripcord”

Not only do these images evoke similarity to each other but we also find them strikingly similar

to a well-studied fractal   “Cantor  Dust” (Figure 5.29) [40, 56]. Cantor Dust is created by taking

the cross product of cantor sets.

Figure 5.29: Cantor dust [55]

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Given the   image  above   it’s  easy   to  retrospectively  go  back  and   find  ways   to  match   the   lyrical  

repetition to segments of the dust.

5.3 Measuring Fractals in Lyrical Verse

One of the most interesting and essential features of fractals is determining their

dimensionality. In the previous sections we have shown clear evidence that the structure of

repetition in dense lyrical verse frequently produces objects with fractal-like properties. In the

general case we can think of the fractal dimension as the way that m repetitions of a pattern

scale by a factor of r [14, 26, 40].

The standard way of measuring fractal is using the Hausdorff dimension described in the

following formula [6, 14, 24, 40]:

D  :=   lim→

log  Nlog(1/r)

Generally we can consider the Hausdorff dimensions as describing the rate that the length of

the fractal, N, scales exponentially with r. The Hausdorff dimension is particularly useful when

dealing with mathematically defined fractals (see chapter 2 for details).

Since we’re  dealing  with  fractals observed in nature, which are discrete, and their scale

is bounded, we are going to have to rely on an empirical method of determining their fractal

dimension. For our purposes we are going to use the mass dimension [40] which asserts that

for mass M and radius R, with a dimension D:

M ≈  R

This in turn means that:

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D   ≈  log(M)log(R)

We can empirically calculate D by observing how M changes as R increases. We can see three

examples of calculating the mass dimension of the observed fractals in Figure 5.30.

Figure 5.30 Computing the mass dimension of (a) Tame  Impala’s  “It  is  not  meant  to  be” (b) Vampire  Weekend’s  “Cousins”  and  (c) Lana  Del  Rey’s  “Radio”.  On  top  is  the  visualized  

expanding value of R, and the bottom we see the log-log plot of M to R

In the Figures displayed [9, 18, 34] we see a visual representation of the radius R expanding

around the fractal component of the lyrics. Below this visualization is then the log-log plot of M

and R. By using linear regression we are able to estimate the power law that is driving the

fractal growth.

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5.4 Interpreting the Fractal Dimension of Lyrics

At this point an obvious question to ask is, "What does it mean for lyrical repetition to

have 1.38 dimensions?" The answer to this question is actually at the heart of why these fractals

are interesting and useful in the first place. Non-integer dimensions typically embody the fact

that while a fractal may be embed in a specific dimension, or resemble one; it has properties

that seem counter intuitive. Take for example Mandelbrot's initial paper on measuring the coast

of Britain. The basic idea is that if we start with a measuring stick of say size 'n' (i.e. 1 meter) we

find one length for the coast, as n decreases we would expect the coastline measurement to

converge on a true length of the coast. But because the coastline's dimension is closer to 1.29

this is not actually what happens. Instead we find that the coast continues to expand in length!

To get a better sense of this, we will look at similar dimension calculations for prose. In

this  case  a  segment  taken  from  Jacques  Ellul’s  The Technological Society [13]. In Figure 5.31

we see this representation using the same threshold for similarity that we used for our song

lyrics. As can be seen this finds the passage to be completely without similarity. However this

representation is a bit unfair to the prose since there are plenty of recurring ideas and themes in

this passage, but the sentences are more complicated (and lengthier). To correct this, in Figure

5.32 we see the plot for the reduce threshold which lets us see some of the sentences which do

have   repetition   in   content.   In   this   latter   representation   we   see   that   the   text’s   dimension   is  

approximately 1.

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Figure 5.31 Calculating the mass dimension for a section from The Technological Society using the same threshold for similarity as all of the lyric calculations.

Figure 5.32 Calculating the mass dimension for a section from The Technological Society using a more relaxed threshold.

We can think of the default stream of speech, with no special repetition as a 1

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dimensional line. But songs are special precisely because they are non-linear in the same way.

Exactly what makes lyrics, with repeated choruses and verses so appealing is this extra-

dimensionality. Poetry and especially song lyrics are different from prose in essentially the same

way that the coast of Britain is different from the Nevada/Utah border. In this way by

understanding the fractal nature of lyrical verse we begin to unravel the mathematics of

aesthetics.

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Chapter 6 Conclusion

“In  my  end  is  my  beginning.”

-- T.S. Eliot, The Four Quartets

6.1 Conclusion

In 1933 George Birkhoff, primarily known for his work on Ergodic theory, a field of

dynamical systems, wrote what has become a relatively obscure work titled Aesthetic Measure

[4]. In this work Birkhoff attempts to find a mathematical model of aesthetics. His initial model

determines that the aesthetic measure M  is composed of two main components order O  and

complexity 𝐶  and that aesthetic measure can be understood with this simple formula:

M   =  OC

To really understand what Birkhoff is getting at it is essential to expand on what is meant by

‘order’  and   ‘complexity’.  Complexity   relates   to   the  “feeling  of   effort”   in  perceiving   the  aesthetic  

experience. For example a harmony with a very few distinct notes requires very little effort on

behalf of the listener, the more complex the music becomes the more tension is created for the

listener. For defining order Birkhoff  lists  a  series  of  element  which  he  considers  to  be  ‘positive’:  

repetition, similarity, contrast, equality, symmetry, balance, and sequence; and a series of

element  he  considers  ‘negative’:  ambiguity,  undue  repetition,  and  unnecessary  imperfection.  He

points out the jarring experience of a single wrong note appearing the middle of a performance

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and  an  example  of  ‘unnecessary  imperfection’,  and  an  overall  example  of  when  order  is  lacking  

in the aesthetic experience.

While there is little doubt that much  nuance  is  missing  from  Birkhoff’s  analysis, there is

an essential point which is useful in understanding the results from visualizing the structure of

lyrical verse. Aesthetics is a balancing act of trying to create order without overly taxing the

attention of the observer. This is precisely why our visualizations are essential to understanding

the true aesthetic nature of lyrical verse: the patterns that lead to the aesthetically pleasing

nature of verse must remain hidden from the audience to maximize the balance between order

and complexity.

Perhaps the most important insight gained in our work is not simply that there are fractal

patterns that drive the structure of lyrical verse, but specifically that there are hidden patterns

that need to be coaxed out of the work to be understood. As a counterpoint take the fairly

ubiquitous Mandelbrot set (Figure 6.1).

Figure 6.1: Mandelbrot set [63]

This image is, for many people, the definitive image of what a fractal is. It is certainly

aesthetically appealing,  but  we  would  argue  not  nearly  as  appealing  as  one’s  favorite  poem  or  

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song. And yet we have found what are quite clearly fractal patterns in the lyrical structure of

popular verse, so what makes one fractal more appealing than another? To account for this we

only need to observe the visible complexity of the Mandelbrot set, which possesses a profound

order but at the cost of extreme complexity. The lyrical verses we have visualized only display

this complexity when we intentionally dig it out. The order created by these fractal patterns is

observed, while the aesthetic experience is heightened because the complexity driving this is

hidden from the observer.

The great value of visualizing similarity in lyrical verse is that it brings out the unseen but

essential structure of these works. The lesson for continued mathematical understanding of

aesthetics is to seek the very likely hidden processes that drive a beautifully simple order.

6.2 Future Work

6.2.1 Modeling Other Aspects of Repetition

The most obvious deficit in our current research is that we only focus on a very specific,

relatively small portion of the overall structure of repetition. Our work remains concerned with

only the reuse and repetition of exact words. Other forms of repetition certainly worth exploring

include: repetition of sounds, word meaning and themes, and meter. Approaches to each of

these would pose considerably more difficult challenges, especially ones pertaining to the

meaning of language.

6.2.2 Website for Aggregation

Perhaps the most useful future work would be to simply create a web presence where a

continually large sample of lyric visualization could be stored and explored. Certainly larger

structures of patterns would emerge as a larger and larger collection of visualization was stored.

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Part of such a project could also include tools for the automatic processing and visualization of

user uploaded texts. Creating a community of users interested in lyrical patterns and continually

expanding knowledge in the area would be a tremendous benefit.

Another feature which would be great to add would be user interaction. Being able to

interactively highlight repetition and display which specific text corresponded to which moment

of repetition would likely provide even greater insights that we have explored here.

6.2.3 Generative Models

If the true aim is to gain a deeper understanding of aesthetics, there can be no truer text

that to eventually generate lyrics using models created from the patterns we have collected and

observed. More research would need to be done in the aforementioned forms of repetition not

covered to successfully generate complete texts. However even generating similar forms and

patterns that we have observed could be extremely useful and insightful.

6.2.4 Generalization

While the current work is focused specifically on text and text similarity the work in this

paper can be generalized to virtual any set of ordered data. The generalization works as follows:

Given an ordered set of data (in this case lines of text) which can be represented by a vector

(here rows in a tf-idf matrix) and a similarity function (e.g. cosine similarity) a similarity matrix

can then be trivially constructed and visualized using the exact technique used to visualize

similarity between lines in lyrical verse. This generalization opens the doors to performing

similar analysis on nearly any sequential data, from frames in a motion picture to moves in a

game of chess.

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